Ingeniería Biomédica
2025-09-07
Definition
Any signal that meets any of this conditions \[x\left(t\right)=x\left(t + kT\right)\] \[x\left[n\right]=x\left[t + kT\right]\]
Where \(k\in\mathbb{z}\) and \(T\in\mathbb{R}\)
If \(\( x_1(t) \)\) and \(\( x_2(t) \)\) are periodic with periods \(\( T_1 \)\) and \(\( T_2 \)\):
\[ x_1(t + T_1) = x_1(t), \quad x_2(t + T_2) = x_2(t) \]
The sum of both signals is:
\[ x(t) = x_1(t) + x_2(t) \]
For \(\( x(t) \)\) to be periodic, there must exist a common period \(\( T \)\) such that:
\[ T = k_1 T_1 = k_2 T_2 \]
where ( k_1, k_2 ) are positive integers.
The smallest common period is the least common multiple (lcm) of \(\( T_1 \)\) and \(\( T_2 \)\):
\[ T = \operatorname{lcm}(T_1, T_2) \]
If the ratio of the periods is a rational number:
\[ \frac{T_1}{T_2} \in \mathbb{Q} \]
Then, the sum \(\( x_1(t) + x_2(t) \)\) will be periodic.
If the ratio is irrational, the resulting signal will not be periodic.